Except for the statement pertaining to the distance between slow manifolds in (REF ), statement (i) follows directly from Fenichel theory [1]}, [2]}, [3]}, [4]} and Theorem REF . The distance in (REF ) follows from statement (ii) after extending the local estimate
\(| \varphi _{\varepsilon ,h}(x) - \varphi _{ode,\varepsilon }(x) | = O(\varepsilon ^2 h)\) 
